[253] Plato, Legg. vii. pp. 818 E, 819 B-D. ᾐσχύνθην … ὑπὲρ ἁπάντων τῶν Ἑλλήνων. Compare Legg. v. p. 747 C, and Republic, iv. p. 436 A.
Respecting the distinction between θεοί, δαίμονες, ἥρωες, see Nägelsbach, Nach-Homerische Theologie, pp. 104-115.
[254] Plato, Legg. vii. pp. 819 E, 820 A-C.
[255] Plato, Legg. vii. p. 820 D. μετὰ παιδιᾶς ἅμα μανθανόμενα ὠφελήσει.
I transcribe here the curious passage which we read a little before.
Plat. Legg. vii. p. 819 A-C. Τοσάδε τοίνυν ἕκαστα χρὴ φάναι μανθάνειν δεῖν τοὺς ἐλευθέρους, ὅσα καὶ πάμπολυς ἐν Αἰγύπτῳ παίδων ὄχλος ἅμα γράμμασι μανθάνει. Πρῶτον μὲν γὰρ περὶ λογισμοὺς ἀτεχνῶς παισὶν ἐξευρημένα μαθήματα, μετὰ παιδιᾶς τε καὶ ἡδονῆς μανθάνειν· μήλων τέ τινῶν διανομαὶ καὶ στεφάνων πλείοσιν ἅμα καὶ ἐλάττοσιν, ἁρμοττόντων ἀριθμῶν τῶν αὐτῶν … καὶ δὴ καὶ παίζοντες, φιάλας ἅμα χρυσοῦ καὶ χαλκοῦ καὶ ἀργύρου καὶ τοιούτων τινῶν ἄλλων κεραννύντες, οἱ δὲ καὶ ὅλας πως διαδιδόντες, ὅπερ εἶπον, εἰς παιδιὰν ἐναρμόττοντες τὰς τῶν ἀναγκαίων ἀριθμῶν χρήσεις, ὠφελοῦσι τοὺς μανθάνοντας εἰς τε τὰς τῶν στρατοπέδων τάξεις καὶ ἀγωγὰς καὶ στρατείας καὶ εἰς οἰκονομίας αὖ· καὶ πάντως χρησιμωτέρους αὐτοὺς αὐτοῖς καὶ ἐγρηγορότας μᾶλλον τοὺς ἀνθρώπους ἀπεργάζονται.
The information here given is valuable respecting the extensive teaching of elementary arithmetic as well as of letters among Egyptian boys, far more extensive than among Hellenic boys. The priests especially, in Egypt a numerous order, taught these matters to their own sons (Diodor. i. 81), probably to other boys also. The information is valuable too in another point of view, as respects the method of teaching arithmetic to boys; not by abstract numbers, nor by simple effort of memory in the repetition of a multiplication-table, but by concrete examples and illustrations exhibited to sense in familiar objects. The importance of this concrete method, both in facilitating comprehension and in interesting the youthful learner, are strongly insisted on by Plato, as they have been also by some of the ablest modern teachers of elementary arithmetic: see Professor Leslie’s Philosophy of Arithmetic, and Mr. Horace Grant’s Arithmetic for Young Children and Second Stage of Arithmetic. The following passage from a work of Sir John Herschel (Review of Whewell’s History of Inductive Sciences, in the Quarterly Review, June, 1841) bears a striking and curious analogy to the sentences above transcribed from Plato:— “Number we cannot help regarding as an abstraction, and consequently its general properties or its axioms to be of necessity inductively concluded from the consideration of particular cases. And surely this is the way in which children do acquire their knowledge of number, and in which they learn its axioms. The apples and the marbles are put in requisition (μήλων διανομαὶ καὶ στεφάνων, Plato), and through the multitude of gingerbread nuts their ideas acquire clearness, precision, and generality.”
I borrow the above references from Mr. John Stuart Mill, System of Logic, Book ii. ch. vi. p. 335, ed. 1. They are annexed as a note to the valuable chapters of his work on Demonstration and Necessary Truths, in which he shows that the truth so-called, both in Geometry and Arithmetic, rest upon inductive evidence.
“The fundamental truths of the Science of Number all rest upon the evidence of sense: they are proved by showing to our eyes and to our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers, the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed upon a knowledge of this fact. All who wish to carry the child’s mind along with them in learning arithmetic — all who (as Dr. Biber in his remarkable Letters on Education expresses it) wish to teach numbers and not mere ciphers — now teach it through the evidence of the senses, in the manner we have described” (p. 335).
Astronomy must be taught, in order that the citizens may not assert libellous falsehoods respecting the heavenly bodies.